regression_analysis.ipynb

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    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
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    "language_info": {
      "name": "python"
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  "cells": [
    {
      "cell_type": "markdown",
      "source": [
        "<a href=\"https://colab.research.google.com/github/porterjenkins/CS180/blob/main/data_science_labs/data_science_lab_12.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a><p><b>After clicking the \"Open in Colab\" link, copy the notebook to your own Google Drive before getting started, or it will not save your work</b></p>"
      ],
      "metadata": {
        "id": "IPir_6bCCFnZ"
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    },
    {
      "cell_type": "markdown",
      "source": [
        "# BYU CS 180 Lab 12"
      ],
      "metadata": {
        "id": "_slaQdUGCB0t"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# Dependencies for the lab\n",
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "from sklearn.linear_model import LinearRegression\n",
        "from sklearn.neighbors import KNeighborsRegressor\n",
        "from sklearn.tree import DecisionTreeRegressor\n",
        "import pandas as pd\n",
        "from sklearn.metrics import mean_squared_error\n",
        "import statsmodels.api as sm"
      ],
      "metadata": {
        "id": "Gaema31vjP8f"
      },
      "execution_count": 3,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Introduction:\n",
        "In this lab you will fit regression models on multiple datasets."
      ],
      "metadata": {
        "id": "ct7fnkcnCL8O"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Exercise 1: Synthetic Data\n",
        "First we will fit a regression model to multiple variants of a synthetic dataset. In each case, we know the true parameters (slope and intercept) and we will vary the noise of the data. Execute the following code to generate the data:"
      ],
      "metadata": {
        "id": "PUat5xRAcdrC"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "import numpy as np\n",
        "\n",
        "w_0_true = -2.5 # this is the TRUE intercept\n",
        "w_1_true = 0.1 # this is the TRUE slope\n",
        "\n",
        "def f_x(x, w_0, w_1, sig):\n",
        "  n = len(x)\n",
        "  y = w_1*x + w_0 + np.random.normal(0, sig, n)\n",
        "  return y\n",
        "\n",
        "x = np.arange(0, 100)\n",
        "\n",
        "\n",
        "y_1 = f_x(x, w_0_true, w_1_true, sig=0.5) # standard deviation = 0.5\n",
        "y_2 = f_x(x, w_0_true, w_1_true, sig=1.5) # standard deviation = 1.5\n",
        "y_3 = f_x(x, w_0_true, w_1_true, sig=5.0) # standard deviation = 5\n",
        "y_4 = f_x(x, w_0_true, w_1_true, sig=10.0) # standard deviation = 10"
      ],
      "metadata": {
        "id": "wgyxUegDs57G"
      },
      "execution_count": 2,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "### Question 1a:\n",
        "For each of the four datasets we generated do the following:\n",
        "\n",
        "* Plot a scatter plot of `x` and `y_i` (e.g, `x` and `y_2`)\n",
        "\n",
        "* Fit a [linear regression model](https://scikit-learn.org/stable/modules/generated/sklearn.linear_model.LinearRegression.html) using sklearn (sklearn.linear_model.LinearRegression )\n",
        "\n",
        "  * Make sure `fit_intercept` is set to `True`\n",
        "\n",
        "  * hint: sklearn expects the features to be two-dimensional. You many need to use the following code to reshape your `x` array when training:\n",
        "\n",
        "```\n",
        "  reg = LinearRegression(fit_intercept=True)\n",
        "  reg.fit(x.reshape(-1, 1), y) # reshape x to be two-dimensional\n",
        "```\n",
        "* Print the estimated coefficients:\n",
        "\n",
        "  * These can be accessed with the `coef_` and `intercept_` attributes on the regression object\n",
        "\n",
        "* Plot the regression line on the scatter plots of `x` and `y`\n",
        "\n",
        "Comment on how well the regression model approximates the data"
      ],
      "metadata": {
        "id": "NdAGXZPYtCVW"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "plt.scatter(x, y_1)\n",
        "reg1 = LinearRegression(fit_intercept=True)\n",
        "reg1.fit(x.reshape(-1,1), y_1)\n",
        "print(reg1.coef_)\n",
        "print(reg1.intercept_)\n",
        "results = sm.OLS(y_1, x).fit()\n",
        "slope_hat = results.params[0]\n",
        "x_grid = np.linspace(x.min(), x.max())\n",
        "y_hat = x_grid*slope_hat\n",
        "plt.plot(x_grid, y_hat)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 484
        },
        "id": "N02ZRw_yMC2Y",
        "outputId": "4373f7c7-8b05-4286-c561-1035ec2c5189"
      },
      "execution_count": 29,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "[0.09918634]\n",
            "-2.536535205908829\n"
          ]
        },
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x7ff920c3d250>]"
            ]
          },
          "metadata": {},
          "execution_count": 29
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 640x480 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Doesn't fit the data super well, there are a lot of outliers on the bottom left. "
      ],
      "metadata": {
        "id": "MaJPmHaMiarz"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "plt.scatter(x, y_2)\n",
        "reg2 = LinearRegression(fit_intercept=True)\n",
        "reg2.fit(x.reshape(-1, 1), y_2)\n",
        "print(reg2.coef_)\n",
        "print(reg2.intercept_)\n",
        "results = sm.OLS(y_2, x).fit()\n",
        "slope_hat = results.params[0]\n",
        "x_grid = np.linspace(x.min(), x.max())\n",
        "y_hat = x_grid*slope_hat\n",
        "plt.plot(x_grid, y_hat)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 484
        },
        "id": "tarv7uRweMO1",
        "outputId": "b6bbbbe8-70ae-40c2-db99-001ea7d8d2dc"
      },
      "execution_count": 30,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "[0.09716908]\n",
            "-2.540408747673234\n"
          ]
        },
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x7ff920cac7c0>]"
            ]
          },
          "metadata": {},
          "execution_count": 30
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 640x480 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Still outliers on the bottom left and top right"
      ],
      "metadata": {
        "id": "TqY2OFrFih-e"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "plt.scatter(x, y_3)\n",
        "reg3 = LinearRegression(fit_intercept=True)\n",
        "reg3.fit(x.reshape(-1, 1), y_3)\n",
        "print(reg3.coef_)\n",
        "print(reg3.intercept_)\n",
        "results = sm.OLS(y_3, x).fit()\n",
        "slope_hat = results.params[0]\n",
        "x_grid = np.linspace(x.min(), x.max())\n",
        "y_hat = x_grid*slope_hat\n",
        "plt.plot(x_grid, y_hat)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 484
        },
        "id": "d2NnZiTeekgv",
        "outputId": "538b8941-327b-4dbf-884f-9f36854bbfa4"
      },
      "execution_count": 31,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "[0.10690438]\n",
            "-2.154302950221789\n"
          ]
        },
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x7ff920d3e460>]"
            ]
          },
          "metadata": {},
          "execution_count": 31
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 640x480 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "This data shows a weak linear relationship, so the model can't represent it too well."
      ],
      "metadata": {
        "id": "AtkydmKsilyU"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "plt.scatter(x, y_4)\n",
        "reg4 = LinearRegression(fit_intercept=True)\n",
        "reg4.fit(x.reshape(-1, 1), y_4)\n",
        "print(reg4.coef_)\n",
        "print(reg4.intercept_)\n",
        "results = sm.OLS(y_4, x).fit()\n",
        "slope_hat = results.params[0]\n",
        "x_grid = np.linspace(x.min(), x.max())\n",
        "y_hat = x_grid*slope_hat\n",
        "plt.plot(x_grid, y_hat)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 484
        },
        "id": "r5VBRICqeoLg",
        "outputId": "e12ddfab-15dd-428d-a7c7-f1f1b2b8a3f0"
      },
      "execution_count": 32,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "[0.07176042]\n",
            "-0.7880191967244286\n"
          ]
        },
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x7ff920f9bdf0>]"
            ]
          },
          "metadata": {},
          "execution_count": 32
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 640x480 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "This data shows an even weaker linear relationship so the model does a poor job of representing the data. "
      ],
      "metadata": {
        "id": "VkLyXbPjis2Y"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "### Question 1b:\n",
        "\n",
        "What do you observe as we add noise to our generated dataset? How well are we able to recover the true parameters, `w_0_true` and `w_1_true` from the data as noise increases?"
      ],
      "metadata": {
        "id": "ezTGk3qzw6qC"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "As noise is added, the data moves further and further away from the model. We can recover the true parameters by increasing the accuracy of np.linspace. "
      ],
      "metadata": {
        "id": "JD4s8NkhxCsF"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "# Exercise 2\n",
        "Read the California Housing Data from the `sample_data` on every collab instance:"
      ],
      "metadata": {
        "id": "2haBwY12xHdB"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "import pandas as pd\n",
        "train = pd.read_csv(\"sample_data/california_housing_train.csv\")\n",
        "test = pd.read_csv(\"sample_data/california_housing_test.csv\")"
      ],
      "metadata": {
        "id": "DoL6tiKMxpn7"
      },
      "execution_count": 2,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "This dataset contains features on census tracts in California in the early 1990’s. The target is the variable `median_house_value` of each census tract."
      ],
      "metadata": {
        "id": "zEiUroYyxtlC"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "### Exercise 2a:\n",
        "Plot a [scatter_plot matrix](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.plotting.scatter_matrix.html) using the training data. You may need to tune the `fig_size` argument to make the figure larger. "
      ],
      "metadata": {
        "id": "mcEiKgcxxwyw"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "pd.plotting.scatter_matrix(train, figsize = (12,12))"
      ],
      "metadata": {
        "id": "EieD2s8byLgt"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "What variables appear to be correlated with `median_house_value`?"
      ],
      "metadata": {
        "id": "Qomi7e6yyEEh"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "> median_income, lat and long coords"
      ],
      "metadata": {
        "id": "RtFISgwsyNyY"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "What variables are not correlated with `median_house_value`?"
      ],
      "metadata": {
        "id": "soyZfoFdyF3G"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "> housing_med_age, total_rooms, total_bedrooms, pop, households, "
      ],
      "metadata": {
        "id": "okhRuaEtyRKg"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "What features appear to be correlated with each other?"
      ],
      "metadata": {
        "id": "TQW2x9qByIUY"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "> total_rooms with households, total_bedrooms and population\n",
        "lat and long\n",
        "total_bedrooms with household\n"
      ],
      "metadata": {
        "id": "VvGuKTHXyYe_"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "### Exercise 2b\n",
        "Fit a linear regression model using `sklearn`.\n",
        "* Fit a few different models (each with different features).\n",
        "* Evaluate the [mean_squared_error](https://scikit-learn.org/stable/modules/model_evaluation.html#mean-squared-error) of your models with both the training and test set."
      ],
      "metadata": {
        "id": "aJ2gcLdQyi7M"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "feat = ['latitude', 'longitude', 'median_income']\n",
        "X_train = train[feat]\n",
        "y_train = train[\"median_house_value\"]\n",
        "X_test = test[feat]\n",
        "y_test = test[\"median_house_value\"]\n",
        "model = LinearRegression().fit(X_train, y_train)\n",
        "y_pred = model.predict(X_test)\n",
        "x_pred = model.predict(X_train)\n",
        "mse_y = mean_squared_error(y_test.values, y_pred)\n",
        "mse_x = mse = mean_squared_error(y_train.values, x_pred)\n",
        "print(f'test: {mse_y}, train: {mse_x}')"
      ],
      "metadata": {
        "id": "qfm8NUxsy3WD",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "5aecbb68-18a8-4ea7-ffea-9fc0e2463c92"
      },
      "execution_count": 7,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "test: 5520546528.355815, train: 5535438821.879477\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "feat2 = ['median_income', 'housing_median_age']\n",
        "X_train = train[feat2]\n",
        "y_train = train[\"median_house_value\"]\n",
        "X_test = test[feat2]\n",
        "y_test = test[\"median_house_value\"]\n",
        "model = LinearRegression().fit(X_train, y_train)\n",
        "y_pred = model.predict(X_test)\n",
        "x_pred = model.predict(X_train)\n",
        "mse_y = mean_squared_error(y_test.values, y_pred)\n",
        "mse_x = mse = mean_squared_error(y_train.values, x_pred)\n",
        "print(f'test: {mse_y}, train: {mse_x}')"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "5BcueBpvn-VH",
        "outputId": "ffdd5bfa-4efd-4f21-b639-7b0f135676a0"
      },
      "execution_count": 8,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "test: 6542597407.100302, train: 6535814191.460647\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "feat3 = ['latitude', 'longitude', 'housing_median_age', 'total_rooms', 'median_income', 'population']\n",
        "X_train = train[feat3]\n",
        "y_train = train[\"median_house_value\"]\n",
        "X_test = test[feat3]\n",
        "y_test = test[\"median_house_value\"]\n",
        "model = LinearRegression().fit(X_train, y_train)\n",
        "y_pred = model.predict(X_test)\n",
        "x_pred = model.predict(X_train)\n",
        "mse_y = mean_squared_error(y_test.values, y_pred)\n",
        "mse_x = mse = mean_squared_error(y_train.values, x_pred)\n",
        "print(f'test: {mse_y}, train: {mse_x}')"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "g5V1FgZ2jtZH",
        "outputId": "9cf86414-1418-4c51-867a-0bf68fa6c9a0"
      },
      "execution_count": 9,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "test: 5138248476.985099, train: 5182741638.603556\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "What did you observe? Which model yields the lowest test error?"
      ],
      "metadata": {
        "id": "NXTMPY6NzShb"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "> the error for all the training data is higher than the test, which can suggest the model is overfitting the data. The model with the lowest error is the third across all variables. "
      ],
      "metadata": {
        "id": "pjIx8Lh9zWH6"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "### Exercise 2c\n",
        "Fit a linear regression using the `statsmodels` api.\n",
        "* Print the model output using `print(results.summary())`"
      ],
      "metadata": {
        "id": "7PsfUy3Izc0Q"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "X = train[[\"longitude\", \"latitude\", \"housing_median_age\", \"total_rooms\", \"total_bedrooms\", \"population\", \"households\", \"median_income\"]]\n",
        "y = train[\"median_house_value\"]\n",
        "result = sm.OLS(y, X).fit()\n",
        "print(result.summary())"
      ],
      "metadata": {
        "id": "aQEoQKtT0F2W",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "fdc50559-e8ca-4a93-f286-426f78464d6a"
      },
      "execution_count": 47,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "                                 OLS Regression Results                                \n",
            "=======================================================================================\n",
            "Dep. Variable:     median_house_value   R-squared (uncentered):                   0.901\n",
            "Model:                            OLS   Adj. R-squared (uncentered):              0.901\n",
            "Method:                 Least Squares   F-statistic:                          1.927e+04\n",
            "Date:                Thu, 13 Apr 2023   Prob (F-statistic):                        0.00\n",
            "Time:                        17:21:21   Log-Likelihood:                     -2.1492e+05\n",
            "No. Observations:               17000   AIC:                                  4.298e+05\n",
            "Df Residuals:                   16992   BIC:                                  4.299e+05\n",
            "Df Model:                           8                                                  \n",
            "Covariance Type:            nonrobust                                                  \n",
            "======================================================================================\n",
            "                         coef    std err          t      P>|t|      [0.025      0.975]\n",
            "--------------------------------------------------------------------------------------\n",
            "longitude          -2242.3547    119.100    -18.828      0.000   -2475.802   -2008.907\n",
            "latitude           -8474.2259    377.129    -22.470      0.000   -9213.438   -7735.013\n",
            "housing_median_age  1786.1485     49.565     36.036      0.000    1688.995    1883.301\n",
            "total_rooms          -15.3597      0.918    -16.723      0.000     -17.160     -13.559\n",
            "total_bedrooms        79.9623      8.246      9.697      0.000      63.799      96.125\n",
            "population           -39.9046      1.277    -31.238      0.000     -42.409     -37.401\n",
            "households           133.1703      8.918     14.932      0.000     115.689     150.651\n",
            "median_income       4.584e+04    381.200    120.249      0.000    4.51e+04    4.66e+04\n",
            "==============================================================================\n",
            "Omnibus:                     3565.084   Durbin-Watson:                   1.154\n",
            "Prob(Omnibus):                  0.000   Jarque-Bera (JB):            12369.583\n",
            "Skew:                           1.040   Prob(JB):                         0.00\n",
            "Kurtosis:                       6.624   Cond. No.                     3.00e+03\n",
            "==============================================================================\n",
            "\n",
            "Notes:\n",
            "[1] R² is computed without centering (uncentered) since the model does not contain a constant.\n",
            "[2] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
            "[3] The condition number is large,  3e+03. This might indicate that there are\n",
            "strong multicollinearity or other numerical problems.\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Comment on which features appear to have the strongest relationship with `median_house_value`?"
      ],
      "metadata": {
        "id": "VfMAhsYlzuBR"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "> The features that seem to have the strongest relationship with median_house_value are median_income, longitude, latitude, and housing_median_age. "
      ],
      "metadata": {
        "id": "e63Ev6eg0NOV"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Write a sentence interpreting the coefficient for `median_income`."
      ],
      "metadata": {
        "id": "2bon361C0WYx"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "> The coefficient for median_income in the linear regression model is 40,310, so for every increase by one in median income, we would expect median_house_value to increase by $40,301. \n",
        "\n",
        "\n"
      ],
      "metadata": {
        "id": "_YFKpYMP0ZKG"
      }
    }
  ]
}